3.231 \(\int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx\)

Optimal. Leaf size=124 \[ -\frac {\sqrt {1-a^2 x^2} \cosh ^{-1}(a x)^2}{x}-\frac {a \sqrt {a x-1} \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a x}}+\frac {a \sqrt {a x-1} \cosh ^{-1}(a x)^2}{\sqrt {1-a x}}-\frac {2 a \sqrt {a x-1} \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt {1-a x}} \]

[Out]

a*arccosh(a*x)^2*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-2*a*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-
1)^(1/2)/(-a*x+1)^(1/2)-a*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*(a*x-1)^(1/2)/(-a*x+1)^(1/2)-arccosh
(a*x)^2*(-a^2*x^2+1)^(1/2)/x

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Rubi [A]  time = 0.44, antiderivative size = 174, normalized size of antiderivative = 1.40, number of steps used = 7, number of rules used = 7, integrand size = 24, \(\frac {\text {number of rules}}{\text {integrand size}}\) = 0.292, Rules used = {5798, 5724, 5660, 3718, 2190, 2279, 2391} \[ -\frac {a \sqrt {a x-1} \sqrt {a x+1} \text {PolyLog}\left (2,-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (a x+1) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}+\frac {a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {a x-1} \sqrt {a x+1} \cosh ^{-1}(a x) \log \left (e^{2 \cosh ^{-1}(a x)}+1\right )}{\sqrt {1-a^2 x^2}} \]

Antiderivative was successfully verified.

[In]

Int[ArcCosh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

(a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]^2)/Sqrt[1 - a^2*x^2] - ((1 - a*x)*(1 + a*x)*ArcCosh[a*x]^2)/(x*Sq
rt[1 - a^2*x^2]) - (2*a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*ArcCosh[a*x]*Log[1 + E^(2*ArcCosh[a*x])])/Sqrt[1 - a^2*x^
2] - (a*Sqrt[-1 + a*x]*Sqrt[1 + a*x]*PolyLog[2, -E^(2*ArcCosh[a*x])])/Sqrt[1 - a^2*x^2]

Rule 2190

Int[(((F_)^((g_.)*((e_.) + (f_.)*(x_))))^(n_.)*((c_.) + (d_.)*(x_))^(m_.))/((a_) + (b_.)*((F_)^((g_.)*((e_.) +
(f_.)*(x_))))^(n_.)), x_Symbol] :> Simp[((c + d*x)^m*Log[1 + (b*(F^(g*(e + f*x)))^n)/a])/(b*f*g*n*Log[F]), x]
- Dist[(d*m)/(b*f*g*n*Log[F]), Int[(c + d*x)^(m - 1)*Log[1 + (b*(F^(g*(e + f*x)))^n)/a], x], x] /; FreeQ[{F,
a, b, c, d, e, f, g, n}, x] && IGtQ[m, 0]

Rule 2279

Int[Log[(a_) + (b_.)*((F_)^((e_.)*((c_.) + (d_.)*(x_))))^(n_.)], x_Symbol] :> Dist[1/(d*e*n*Log[F]), Subst[Int
[Log[a + b*x]/x, x], x, (F^(e*(c + d*x)))^n], x] /; FreeQ[{F, a, b, c, d, e, n}, x] && GtQ[a, 0]

Rule 2391

Int[Log[(c_.)*((d_) + (e_.)*(x_)^(n_.))]/(x_), x_Symbol] :> -Simp[PolyLog[2, -(c*e*x^n)]/n, x] /; FreeQ[{c, d,
e, n}, x] && EqQ[c*d, 1]

Rule 3718

Int[((c_.) + (d_.)*(x_))^(m_.)*tan[(e_.) + (Complex[0, fz_])*(f_.)*(x_)], x_Symbol] :> -Simp[(I*(c + d*x)^(m +
1))/(d*(m + 1)), x] + Dist[2*I, Int[((c + d*x)^m*E^(2*(-(I*e) + f*fz*x)))/(1 + E^(2*(-(I*e) + f*fz*x))), x],
x] /; FreeQ[{c, d, e, f, fz}, x] && IGtQ[m, 0]

Rule 5660

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)/(x_), x_Symbol] :> Subst[Int[(a + b*x)^n/Coth[x], x], x, ArcCosh
[c*x]] /; FreeQ[{a, b, c}, x] && IGtQ[n, 0]

Rule 5724

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_)*((d1_) + (e1_.)*(x_))^(p_.)*((d2_) + (e2_.)*(x
_))^(p_.), x_Symbol] :> Simp[((f*x)^(m + 1)*(d1 + e1*x)^(p + 1)*(d2 + e2*x)^(p + 1)*(a + b*ArcCosh[c*x])^n)/(d
1*d2*f*(m + 1)), x] + Dist[(b*c*n*(-(d1*d2))^IntPart[p]*(d1 + e1*x)^FracPart[p]*(d2 + e2*x)^FracPart[p])/(f*(m
+ 1)*(1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^(m + 1)*(-1 + c^2*x^2)^(p + 1/2)*(a + b*ArcCosh
[c*x])^(n - 1), x], x] /; FreeQ[{a, b, c, d1, e1, d2, e2, f, m, p}, x] && EqQ[e1 - c*d1, 0] && EqQ[e2 + c*d2,
0] && GtQ[n, 0] && EqQ[m + 2*p + 3, 0] && NeQ[m, -1] && IntegerQ[p + 1/2]

Rule 5798

Int[((a_.) + ArcCosh[(c_.)*(x_)]*(b_.))^(n_.)*((f_.)*(x_))^(m_.)*((d_) + (e_.)*(x_)^2)^(p_), x_Symbol] :> Dist
[((-d)^IntPart[p]*(d + e*x^2)^FracPart[p])/((1 + c*x)^FracPart[p]*(-1 + c*x)^FracPart[p]), Int[(f*x)^m*(1 + c*
x)^p*(-1 + c*x)^p*(a + b*ArcCosh[c*x])^n, x], x] /; FreeQ[{a, b, c, d, e, f, m, n, p}, x] && EqQ[c^2*d + e, 0]
&&  !IntegerQ[p]

Rubi steps

\begin {align*} \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {1-a^2 x^2}} \, dx &=\frac {\left (\sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)^2}{x^2 \sqrt {-1+a x} \sqrt {1+a x}} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \int \frac {\cosh ^{-1}(a x)}{x} \, dx}{\sqrt {1-a^2 x^2}}\\ &=-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int x \tanh (x) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {\left (4 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {e^{2 x} x}{1+e^{2 x}} \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (2 a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \log \left (1+e^{2 x}\right ) \, dx,x,\cosh ^{-1}(a x)\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}+\frac {\left (a \sqrt {-1+a x} \sqrt {1+a x}\right ) \operatorname {Subst}\left (\int \frac {\log (1+x)}{x} \, dx,x,e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ &=\frac {a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x)^2}{\sqrt {1-a^2 x^2}}-\frac {(1-a x) (1+a x) \cosh ^{-1}(a x)^2}{x \sqrt {1-a^2 x^2}}-\frac {2 a \sqrt {-1+a x} \sqrt {1+a x} \cosh ^{-1}(a x) \log \left (1+e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}-\frac {a \sqrt {-1+a x} \sqrt {1+a x} \text {Li}_2\left (-e^{2 \cosh ^{-1}(a x)}\right )}{\sqrt {1-a^2 x^2}}\\ \end {align*}

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Mathematica [A]  time = 0.48, size = 111, normalized size = 0.90 \[ \frac {a \sqrt {\frac {a x-1}{a x+1}} (a x+1) \left (\text {Li}_2\left (-e^{-2 \cosh ^{-1}(a x)}\right )+\cosh ^{-1}(a x) \left (\frac {\sqrt {\frac {a x-1}{a x+1}} (a x+1) \cosh ^{-1}(a x)}{a x}-\cosh ^{-1}(a x)-2 \log \left (e^{-2 \cosh ^{-1}(a x)}+1\right )\right )\right )}{\sqrt {-((a x-1) (a x+1))}} \]

Warning: Unable to verify antiderivative.

[In]

Integrate[ArcCosh[a*x]^2/(x^2*Sqrt[1 - a^2*x^2]),x]

[Out]

(a*Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*(ArcCosh[a*x]*(-ArcCosh[a*x] + (Sqrt[(-1 + a*x)/(1 + a*x)]*(1 + a*x)*A
rcCosh[a*x])/(a*x) - 2*Log[1 + E^(-2*ArcCosh[a*x])]) + PolyLog[2, -E^(-2*ArcCosh[a*x])]))/Sqrt[-((-1 + a*x)*(1
+ a*x))]

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fricas [F]  time = 0.54, size = 0, normalized size = 0.00 \[ {\rm integral}\left (-\frac {\sqrt {-a^{2} x^{2} + 1} \operatorname {arcosh}\left (a x\right )^{2}}{a^{2} x^{4} - x^{2}}, x\right ) \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="fricas")

[Out]

integral(-sqrt(-a^2*x^2 + 1)*arccosh(a*x)^2/(a^2*x^4 - x^2), x)

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giac [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {arcosh}\left (a x\right )^{2}}{\sqrt {-a^{2} x^{2} + 1} x^{2}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="giac")

[Out]

integrate(arccosh(a*x)^2/(sqrt(-a^2*x^2 + 1)*x^2), x)

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maple [A]  time = 0.36, size = 241, normalized size = 1.94 \[ -\frac {\sqrt {-a^{2} x^{2}+1}\, \left (a^{2} x^{2}-\sqrt {a x +1}\, \sqrt {a x -1}\, a x -1\right ) \mathrm {arccosh}\left (a x \right )^{2}}{x \left (a^{2} x^{2}-1\right )}-\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right )^{2} a}{a^{2} x^{2}-1}+\frac {2 \sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \mathrm {arccosh}\left (a x \right ) \ln \left (1+\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1}+\frac {\sqrt {-a^{2} x^{2}+1}\, \sqrt {a x -1}\, \sqrt {a x +1}\, \polylog \left (2, -\left (a x +\sqrt {a x -1}\, \sqrt {a x +1}\right )^{2}\right ) a}{a^{2} x^{2}-1} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x)

[Out]

-(-a^2*x^2+1)^(1/2)*(a^2*x^2-(a*x+1)^(1/2)*(a*x-1)^(1/2)*a*x-1)*arccosh(a*x)^2/x/(a^2*x^2-1)-2*(-a^2*x^2+1)^(1
/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/(a^2*x^2-1)*arccosh(a*x)^2*a+2*(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^(1/2)/
(a^2*x^2-1)*arccosh(a*x)*ln(1+(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*a+(-a^2*x^2+1)^(1/2)*(a*x-1)^(1/2)*(a*x+1)^
(1/2)/(a^2*x^2-1)*polylog(2,-(a*x+(a*x-1)^(1/2)*(a*x+1)^(1/2))^2)*a

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maxima [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \frac {{\left (a^{2} x^{2} - 1\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )^{2}}{\sqrt {a x + 1} \sqrt {-a x + 1} x} - \int \frac {2 \, {\left (a^{3} x^{2} + \sqrt {a x + 1} \sqrt {a x - 1} a^{2} x - a\right )} \log \left (a x + \sqrt {a x + 1} \sqrt {a x - 1}\right )}{{\left (\sqrt {a x + 1} a x^{2} + {\left (a x + 1\right )} \sqrt {a x - 1} x\right )} \sqrt {-a x + 1}}\,{d x} \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(arccosh(a*x)^2/x^2/(-a^2*x^2+1)^(1/2),x, algorithm="maxima")

[Out]

(a^2*x^2 - 1)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))^2/(sqrt(a*x + 1)*sqrt(-a*x + 1)*x) - integrate(2*(a^3*x^2
+ sqrt(a*x + 1)*sqrt(a*x - 1)*a^2*x - a)*log(a*x + sqrt(a*x + 1)*sqrt(a*x - 1))/((sqrt(a*x + 1)*a*x^2 + (a*x
+ 1)*sqrt(a*x - 1)*x)*sqrt(-a*x + 1)), x)

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mupad [F]  time = 0.00, size = -1, normalized size = -0.01 \[ \int \frac {{\mathrm {acosh}\left (a\,x\right )}^2}{x^2\,\sqrt {1-a^2\,x^2}} \,d x \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

int(acosh(a*x)^2/(x^2*(1 - a^2*x^2)^(1/2)),x)

[Out]

int(acosh(a*x)^2/(x^2*(1 - a^2*x^2)^(1/2)), x)

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sympy [F]  time = 0.00, size = 0, normalized size = 0.00 \[ \int \frac {\operatorname {acosh}^{2}{\left (a x \right )}}{x^{2} \sqrt {- \left (a x - 1\right ) \left (a x + 1\right )}}\, dx \]

Verification of antiderivative is not currently implemented for this CAS.

[In]

integrate(acosh(a*x)**2/x**2/(-a**2*x**2+1)**(1/2),x)

[Out]

Integral(acosh(a*x)**2/(x**2*sqrt(-(a*x - 1)*(a*x + 1))), x)

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